Find f(g(x)) f(x)=x^3+12x^2+25x-70 g(x)=x+7

$f (x) = x^{3} + 12 x^{2} + 25 x - 70$ $g (x) = x + 7$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x + 7)$ by substituting in the value of $g$ into $f$ .

$f (x + 7) = {(x + 7)}^{3} + 12 {(x + 7)}^{2} + 25 (x + 7) - 70$

Step 3

Simplify each term.

Use the Binomial Theorem.

$f (x + 7) = x^{3} + 3 x^{2} \cdot 7 + 3 x \cdot 7^{2} + 7^{3} + 12 {(x + 7)}^{2} + 25 (x + 7) - 70$

Simplify each term.

Multiply $7$ by $3$ .

$f (x + 7) = x^{3} + 21 x^{2} + 3 x \cdot 7^{2} + 7^{3} + 12 {(x + 7)}^{2} + 25 (x + 7) - 70$

Raise $7$ to the power of $2$ .

$f (x + 7) = x^{3} + 21 x^{2} + 3 x \cdot 49 + 7^{3} + 12 {(x + 7)}^{2} + 25 (x + 7) - 70$

Multiply $49$ by $3$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 7^{3} + 12 {(x + 7)}^{2} + 25 (x + 7) - 70$

Raise $7$ to the power of $3$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 {(x + 7)}^{2} + 25 (x + 7) - 70$

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 ((x + 7) (x + 7)) + 25 (x + 7) - 70$

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 (x (x + 7) + 7 (x + 7)) + 25 (x + 7) - 70$

Apply the distributive property.

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 (x \cdot x + x \cdot 7 + 7 (x + 7)) + 25 (x + 7) - 70$

Apply the distributive property.

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 (x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7) + 25 (x + 7) - 70$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 (x^{2} + x \cdot 7 + 7 x + 7 \cdot 7) + 25 (x + 7) - 70$

Move $7$ to the left of $x$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 (x^{2} + 7 \cdot x + 7 x + 7 \cdot 7) + 25 (x + 7) - 70$

Multiply $7$ by $7$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 (x^{2} + 7 x + 7 x + 49) + 25 (x + 7) - 70$

Add $7 x$ and $7 x$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 (x^{2} + 14 x + 49) + 25 (x + 7) - 70$

Apply the distributive property.

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 x^{2} + 12 (14 x) + 12 \cdot 49 + 25 (x + 7) - 70$

Simplify.

Multiply $14$ by $12$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 x^{2} + 168 x + 12 \cdot 49 + 25 (x + 7) - 70$

Multiply $12$ by $49$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 x^{2} + 168 x + 588 + 25 (x + 7) - 70$

Apply the distributive property.

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 x^{2} + 168 x + 588 + 25 x + 25 \cdot 7 - 70$

Multiply $25$ by $7$ .

$f (x + 7) = x^{3} + 21 x^{2} + 147 x + 343 + 12 x^{2} + 168 x + 588 + 25 x + 175 - 70$

Step 4

Simplify by adding terms.

Add $21 x^{2}$ and $12 x^{2}$ .

$f (x + 7) = x^{3} + 33 x^{2} + 147 x + 343 + 168 x + 588 + 25 x + 175 - 70$

Add $147 x$ and $168 x$ .

$f (x + 7) = x^{3} + 33 x^{2} + 315 x + 343 + 588 + 25 x + 175 - 70$

Add $315 x$ and $25 x$ .

$f (x + 7) = x^{3} + 33 x^{2} + 340 x + 343 + 588 + 175 - 70$

Simplify by adding and subtracting.

Add $343$ and $588$ .

$f (x + 7) = x^{3} + 33 x^{2} + 340 x + 931 + 175 - 70$

Add $931$ and $175$ .

$f (x + 7) = x^{3} + 33 x^{2} + 340 x + 1106 - 70$

Subtract $70$ from $1106$ .

$f (x + 7) = x^{3} + 33 x^{2} + 340 x + 1036$

Find f(g(x)) f(x)=x^3+12x^2+25x-70 g(x)=x+7

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